Abstract
We give a method for approximating any n-dimensional lattice with a lattice Λ whose factor group ℤn/ ∧ has n-1 cycles of equal length with arbitrary precision. We also show that a direct consequence of this is that the Shortest Vector Problem and the Closest Vector Problem cannot be easier for this type of lattices than for general lattices.
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References
Ajtai, M.: Generating Hard Instances of Lattice Problems. In: Proc. 28th ACM Symposium on Theory of Computing, pp. 99–108 (1996)
Ajtai, M.: The shortest vector problem in I 2 is NP-hard for randomized reductions. In: Proc. 30th ACM Symposium on the Theory of Computing, pp. 10–19 (1998)
Cai, J.-Y., Nerurkar, A.: An Improved Worst-Case to Average-Case Connection for Lattice Problems. In: Proc. 38th IEEE Symposium on Foundations of Computer Science, pp. 468–477 (1997)
Dinur, I.: Approximating SVP∞ to within almost polynomial factors is NP-hard. CIAC 2000. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 263–276. Springer, Heidelberg (2000)
Goldreich, O., Goldwasser, S.: On the limits of non-approximability of lattice problems. Journal of Computer and System Sciences 60(3), 540–563 (2000), Can be obtained from http://www.eccc.uni-trier.de/eccc
Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)
Kannan, R., Bachem, A.: Polynomial Algorithms for Computing of the Smith and Hermite Normal Forms of an Integer Matrix. SIAM Journal of Computing 8, 499–507 (1979)
Khot, S.: Hardness of Approximating the Shortest Vector Problem in High Lp Norms. In: Proc. 44th IEEE Symposium on Foundations of Computer Science, pp. 290–297 (2003)
Lagarias, J.C.: The Computational Complexity of Simultaneous Diophantine Approximation Problems. SIAM Journal of Computing 14, 196–209 (1985)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring Polynomials with Rational Coefficients. Mathematische Annalen 261, 515–534 (1982)
Micciancio, D.: Improving Lattice Based Cryptosystems Using the Hermite Normal Form. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 126–145. Springer, Heidelberg (2001)
Micciancio, D.: The Shortest Vector in a Lattice is Hard to Approximate within Some Constant. SIAM Journal of Computing 30, 2008–2035 (2001)
Paz, A., Schnorr, C.P.: Approximating Integer Lattices by Lattices with Cyclic Lattice Groups. Automata, languages and programming (Karlsruhe), pp. 386–393 (1987)
Trolin, M.: The Shortest Vector Problem in Lattices with Many Cycles. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 194–205. Springer, Heidelberg (2001)
Trolin, M.: Lattices with Many Cycles are Dense (full version), Can be obtained from http://www.nada.kth.se/~marten
van Emde Boas, P.: Another NP-complete partition problem and the complexity of computing short vectors in lattices. Technical Report 81-04. Mathematics Department, University of Amsterdam (1981), Can be obtained from http://turing.wins.uva.nl/~peter
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Trolin, M. (2004). Lattices with Many Cycles Are Dense. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_33
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DOI: https://doi.org/10.1007/978-3-540-24749-4_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21236-2
Online ISBN: 978-3-540-24749-4
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